External review, round 1 · reviews v1 · verdict: major errors

# External verification: 02-two-free-bodies

- **Subproject:** 07-ilang-weight
- **Package:** 02-two-free-bodies
- **Verified version:** v1
- **External round:** 1 of 3
- **Date:** 2026-10-20T22:59:02+02:00
- **Focus points:** none

---
VERDICT: major errors

## Summary

The lowest-state held-body calculation is sound: the exact separation and exchange symmetries, the second- and third-order mutual terms, and the failure of factorization are supported by the supplied code. The free-body analysis has substantial problems, however: it identifies a winding current with the derivative of a position operator on a finite ring, and it uses labeled states and observables that are inadmissible for identical-type bodies. The requested highest-state treatment on odd rings is also explicitly left unfinished. Additional issues concern conserved quantities, packet-width approximations, and the isolation conditions used in the numerical checks.

## Issues

### I1. Ring current is not the derivative of the stated mean position
- **Location:** Steps 5–6, Eqs. (2.12)–(2.13); check_mutual_fall.py, body_ops and run_case
- **Severity:** major
- **Problem:** The derivation uses \([x_1,S_\pm]=\pm S_\pm\), treating every forward ring hop as a displacement of \(+1\). This is false for the diagonal coordinate operator \(X=\sum_{x=0}^{L-1}xP_x\): for example,
  \[
  [X,T]=T-L|0\rangle\langle L-1|.
  \]
  Consequently, the double-commutator calculation omits boundary terms. The code likewise assigns displacement \(+1\) to every forward bond, including the wrapping bond; its velocity is a winding-current operator, not \(i[C,X]\). Its evolution check therefore validates the acceleration of that current, not the claimed second derivative of the mean position. Arbitrary real envelopes in R3 do not justify neglecting the distinction.
- **Suggested fix:** Define the drift observable operationally and derive its relation to mean-position motion. If using a local coordinate chart, restrict the starts and evolution interval appropriately and bound the boundary-tail errors. Make the numerical observable match that definition, and include the corresponding finite-ring errors in (2.12)–(2.13).

### I2. Identical-type bodies are treated as labeled objects
- **Location:** Setup R3; Step 1; Steps 4–6; Consistency check 3
- **Severity:** major
- **Problem:** The stated class includes bodies of the same type, but the vectors \(e_{jl}\) and the product-envelope start are generally not in the required swap-invariant or swap-sign-changing subspace. For such bodies, \(x_1\), \(x_2\), and the individual \(D_i\) are also not label-invariant observables; in general they do not even preserve the admissible subspace. A mirror-symmetric pair of labeled envelopes is not, by itself, a symmetrized or antisymmetrized state. Thus the claimed individual accelerations and conservation statements are not well defined throughout the stated class, contrary to M6 and A6.
- **Suggested fix:** Perform the same-type calculation in the admissible family subspace and specify label-independent observables identifying the two spatial packets, where possible. Distinguishable types may be used for the labeled calculation, but the requested same-type cases require a separate admissible treatment.

### I3. Highest states on odd rings remain unanswered
- **Location:** Step 2; Step 3; Directions in Step 6; Open issues
- **Severity:** major
- **Problem:** The question’s scope includes every \(L\ge5\), and item 3 requests the branch-dependent directions for both lowest and highest field states. The derivation handles highest states only for even \(L\), explicitly leaving odd \(L\) open. On odd rings the uncoupled highest level is degenerate, so neither the lowest-state perturbation calculation nor the even-ring spectral inversion settles the required eigenvalue branch, start dependence, leading mutual interaction, or acceleration directions. Acknowledging the degeneracy does not answer these requested cases.
- **Suggested fix:** Treat the degenerate highest eigenspace on odd rings, specify the admissible selected starts or branches, and derive the requested held interaction and free-body conclusions with their validity conditions.

### I4. An exact body-only conserved quantity is omitted
- **Location:** Step 1 and the conserved-quantities Result bullet
- **Severity:** minor
- **Problem:** For even \(L\), each distinguishable body has the additional exact conserved operator
  \[
  Q_i=\left(\sum_x(-1)^xP_x^{(i)}\right)\sigma_3^{(i)}.
  \]
  Position parity and \(\sigma_3\) each anticommute with the motion hopping, so their product commutes with it, including the field-dependent bond factors of coupling (c). It also commutes with the rest coupling and the field contract. The answer concerning quantities conserved by the bodies alone therefore omits a nontrivial conserved quantity built directly from their places.
- **Suggested fix:** Include these conserved operators and their consequences. For identical-type bodies, state the corresponding permutation-invariant conserved combinations rather than labeled individual operators.

### I5. The correction evaluated at the mean separation needs another approximation
- **Location:** Step 7, the ratio formula following Eq. (2.16)
- **Severity:** minor
- **Problem:** Replacing the correction inside the acceleration sums by \(\bar M^{(2)}(\bar d)\) is not justified merely by calling the packets “broad.” For equal Gaussian widths, the correction generally involves a force-weighted average of the smoothed mutual term,
  \[
  \frac{\sum_{j,l}p_{jl}\,\bar M^{(2)}(l-j)\,\mathcal M'(l-j)}
       {\sum_{j,l}p_{jl}\,\mathcal M'(l-j)},
  \]
  not its value at the mean separation. Broad packets suppress lattice-envelope effects but need not be narrow relative to the spatial variation of the interaction. No additional scale hierarchy or error estimate is supplied.
- **Suggested fix:** Retain the weighted-average expression, or state a packet-size versus interaction-variation condition and estimate the error in replacing it by a value at \(\bar d\).

### I6. The isolation conditions used analytically and numerically are incomplete
- **Location:** Setup R1; Step 4, Eq. (2.11); Code summary for check_mutual_fall.py
- **Severity:** minor
- **Problem:** Joint two-body isolation in R1 does not by itself establish isolation of the single-body levels used to define \(\bar\kappa\) and the inertias. That additional condition is needed when importing (6.10). Moreover, the numerical claim “R1 holds: \(2M_i-4t_f\ge1.2\)” checks only a branch-energy separation, not the gap from the selected group to all other levels. Same-branch field excitations also matter: the uncoupled field gap for the numerical ring is \(4t_f\sin^2(\pi/14)\simeq0.099\), comparable to or smaller than the largest motion amplitudes used. The overlap-based band-selection assertions do not establish the stated \(\tau_i\ll\delta\) condition.
- **Suggested fix:** State the required single-body isolation assumptions explicitly. Compute and report the full relevant two-body and single-body spectral gaps for the numerical cases, and compare the motion-operator norms with those gaps; otherwise qualify which runs are outside the stated asymptotic regime.

## Focus points

None given.

Author: Ivan Honis · ivan.honis@ndot.io · License: CC BY 4.0 · Human–AI note